Lathisms Lecture: Generalized Parking Function Polytopes скачать в хорошем качестве

Lathisms Lecture: Generalized Parking Function Polytopes

Abstract: A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of length $n$ is an $n$-dimensional polytope in $\mathbb{R}^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-parking function polytopes.This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary. Dr. Andrés R. Vindas Meléndez is a Costa Rican-American mathematician, raised in Lynwood, South East Los Angeles, California. He is a first-generation college graduate and is currently a National Science Foundation Postdoctoral Fellow at the University of California, Berkeley and will start as a tenure-track assistant professor at Harvey Mudd College in July 2024. He completed his PhD at the University of Kentucky where he was supported by a National Science Foundation Graduate Research Fellowship and by a National Science Foundation Louis Stokes Alliance for Minority Participation Bridge to Doctorate Fellowship. At the University of Kentucky he was also an affiliated graduate student in the Latin American Studies program and earned a graduate certificate in Latin American, Caribbean, and Latino/a Studies. He earned a master’s degree in mathematics at San Francisco State University and completed his undergraduate degree in mathematics at the University of California, Berkeley where he also minored in Philosophy and Chicana/o & Latina/o Studies. His research interests are in algebraic, enumerative, and geometric combinatorics. Dr. Vindas Meléndez’s teaching, service, mentorship, and outreach is student-centered. He has the opportunity to help guide students to learn abstract mathematics and find their voice while also developing a sense of ownership of their knowledge and mathematical abilities. Dr. Vindas Meléndez strives to create a community in order to build students’ confidence in spite of society’s negative messages and stigma about mathematics. He also aims to build meaningful and empowering experiences with mathematics, while also challenging others to think about the power structures that are present in and outside mathematical spaces.